Community Detection in Random Networks
This work addresses the fundamental problem of community detection in networks for researchers in statistics and network science, but it is incremental as it builds on existing testing frameworks.
The paper tackles the problem of detecting a dense community in a random network by formalizing it as a hypothesis test, deriving a detection lower bound, and proposing a test that achieves this bound for both known and unknown parameters.
We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1 > p0. We derive a detection lower bound for detecting such a subgraph in terms of N, n, p0, p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomial-time. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem. Our focus in this paper is in the quasi-normal regime where n p0 is either bounded away from zero, or tends to zero slowly.